In mathematics, specifically
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a descendant tree is a
hierarchical structure
A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important ...
that visualizes parent-descendant relations between
isomorphism class
In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.
Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the stru ...
es of finite groups of prime power order
, for a fixed prime number
and varying integer exponents
. Such groups are briefly called ''finite''
''p-groups''. The ''vertices'' of a
descendant tree are isomorphism classes of finite ''p''-groups.
Additionally to their ''order''
, finite ''p''-groups have two further related invariants, the
''nilpotency class'' and the coclass
. It turned out that descendant trees of a particular kind, the so-called pruned coclass trees whose infinitely many vertices share a common coclass
, reveal a repeating finite pattern. These two crucial properties of finiteness and periodicity admit a characterization of all members of the tree by finitely many parametrized
presentations
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Present ...
. Consequently, descendant trees play a fundamental role in the classification of finite ''p''-groups. By means of kernels and targets of
Artin transfer homomorphisms, descendant trees can be endowed with additional structure.
An important question is how the descendant tree
can actually be constructed for an assigned starting group which is taken as the root
of the tree. The
''p''-group generation algorithm is a recursive process for constructing the descendant tree of a given finite ''p''-group playing the role of the tree root. This algorithm is implemented in the computational algebra systems
GAP and
Magma
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural sa ...
.
Definitions and terminology
According to M. F. Newman,
[
] there exist several distinct definitions of the parent
of a finite ''p''-group
. The common principle is to form the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of
by a suitable
normal subgroup which can be either
:# the
centre
Center or centre may refer to:
Mathematics
* Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentri ...
of
, whence
is called the ''central quotient'' of
, or
:# the last non-trivial term
of the
lower central series
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centra ...
of
, where
denotes the nilpotency class of
, or
:# the last non-trivial term
of the
lower exponent-''p'' central series of
, where
denotes the exponent-''p'' class of
, or
:# the last non-trivial term
of the
derived series
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group (mathematics), group is the subgroup (mathematics), subgroup generating set of a group, generated by all the commutators of the group.
Th ...
of
, where
denotes the derived length of
.
In each case,
is called an immediate descendant of
and a ''directed edge'' of the tree is defined either by
in the direction of the
canonical projection
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
onto the quotient
or by
in the opposite direction, which is more usual for descendant trees. The former convention is adopted by C. R. Leedham-Green and M. F. Newman,
[
] by M. du Sautoy and D. Segal,
[
] by C. R. Leedham-Green and S. McKay,
[
] and by B. Eick, C. R. Leedham-Green, M. F. Newman and E. A. O'Brien.
[
] The latter definition is used by M. F. Newman,
by M. F. Newman and E. A. O'Brien,
[
] by M. du Sautoy,
[
] and by B. Eick and C. R. Leedham-Green.
[
]
In the following, the direction of the canonical projections is selected for all edges. Then, more generally, a vertex
is a descendant of a vertex
,
and
is an ancestor of
, if either
is equal to
or there is a ''path''
:
, with
,
of directed edges from
to
. The vertices forming the path necessarily coincide with the
''iterated parents'' of
, with
:
:
, with
,
In the most important special case (P2) of parents defined as last non-trivial lower central quotients, they can also be viewed as the successive ''quotients''
''of class''
of
when the nilpotency class of
is given by
:
:
, with
.
Generally, the descendant tree
of a vertex
is the subtree of all descendants of
, starting at the root
. The maximal possible descendant tree
of the trivial group
contains all finite ''p''-groups and is somewhat exceptional, since, for any parent definition (P1–P4), the trivial group
has infinitely many abelian ''p''-groups as its immediate descendants. The parent definitions (P2–P3) have the advantage that any non-trivial finite ''p''-group (of order divisible by
) possesses only finitely many immediate descendants.
Pro-''p'' groups and coclass trees
For a sound understanding of ''coclass trees'' as a particular instance of descendant trees,
it is necessary to summarize some facts concerning infinite
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
pro-''p'' groups.
The members
, with
, of the lower central series of a pro-''p'' group
are closed (and open) subgroups of finite index, and therefore the corresponding quotients
are finite ''p''-groups.
The pro-''p'' group
is said to be of coclass
when the limit
of the coclass of the successive quotients exists and is finite.
An infinite pro-''p'' group
of coclass
is a ''p''-adic pre-
space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it uncha ...
,
since it has a normal subgroup
, the ''translation group'',
which is a free module over the ring
of ''p''-adic integers of ''uniquely determined'' rank
, the ''dimension'',
such that the quotient
is a finite ''p''-group, the ''point group'', which acts on
uniserially.
The dimension is given by
, with some